3.2 Comparison with the Elwert-Haug approximation

In Figure 6.2 we see that the Elwert-Haug results perform fairly well for small momentum transfers but tend to approach the Elwert-Bethe-Heitler results for large momentum transfers. Generally we find that for $d^3\!\sigma$ the Elwert-Haug results seldom overestimate the exact partial wave results by more than 20% (although trends in our results indicate that this overestimate will increase with decreasing incident or outgoing electron energy), while for large momentum transfers the Elwert-Haug results tend to seriously underestimate the exact partial wave results. We find that for large momentum transfers or large emission angles the Elwert-Haug results approach the Bethe-Heitler results within a multiplicative constant which is approximately the Elwert-Factor. We find that for the Elwert-Haug results the large momentum transfer region is roughly defined by $q > (Z\alpha)\,mc$. Integration of the Elwert-Haug results over the outgoing electron angle to obtain $d^2\!\sigma$typically gives a significant underestimate of the exact partial wave results due to the underestimates at large momentum transfers. This is consistent with Fink's findings that the Elwert-Haug results were as poor as the Bethe-Heitler results for $d^2\!\sigma$ for high Zelements, for energies similar to those considered here.